\begin{frame}[allowframebreaks]
\frametitle{Recognizable picture languages}

\begin{thm}
	REC is closed under:
	\begin{itemize}
		\item intersection
		\item union
		\item projection
		\item row and column concatenation and row and column closure operations
		\item projection
		\item rotation
	\end{itemize}
\end{thm}

\begin{proof}
	see \cite{Giammarresi1997}\cite{cherubini2009picture}. 
\end{proof}

\begin{thm}
	REC is not closed under complement. 
\end{thm}

\begin{thm}
	The membership problem for REC is NP-complete
	The emptiness and universe problems for REC are undecidable. 
\end{thm}

\end{frame}